Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions

Abstract

The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({{\mathbb R}}^n;S^m{{\mathbb R}}^n)$ was obtained in [V. Krishnan and V. Sharafutdinov. Range characterization of ray transform on Sobolev spaces of symmetric tensor fields. Inverse Problems and Imaging, 18(6), 1272--1293, 2024]. Here we obtain a range characterization result in higher order weighted Sobolev spaces in two dimensions. Range characterization in the case of $n=2$ is very different from that for $n\ge3$, and this allows us to obtain such a result in higher order weighted Sobolev spaces $H^{r,s}_t(\mathbb{R}^2)$ for any real $r$. Nevertheless, our main tool is again the Reshetnyak formula stating that $\lVert I_mf\rVert_{H^{(r,s+1/2)}_{t+1/2}(T{{\mathbb S}}^{n-1})}=\lVert f\rVert_{H^{(r,s)}_t({{\mathbb R}}^n;S^m{{\mathbb R}}^n)}$ for a solenoidal tensor field $f$.